Degenerate Perturbations of a Two-Phase Transition Model

We study the $\Gamma$-convergence as $\varepsilon \to 0^+$ of the family of degenerate functionals $Q_\varepsilon(u) = \varepsilon \int_\Omega \langle ADu, Du\rangle \, dx + \frac{1}{\varepsilon} \int_\Omega W(u) \, dx$, where A(x) is a symmetric, non negative $n\times n$ matrix on $\Omega$ (i.e. $\langle A(x)\xi, \xi\rangle \ge 0$ for all $x \in \Omega$ and $\xi \in \mathbb{R}^n$) with regular entries and $W : \mathbb{R} \to [0, +\infty)$ is a double well potential having two isolated minimum points. Moreover, under suitable assumptions on the matrix A, we obtain a minimal interface criterion for the $\Gamma$-limit functional exploiting some tools of analysis in Carnot-Caratheodory spaces. We extend some previous results obtained for the non degenerate perturbations $Q_\varepsilon$ in the classical gradient theory of phase transitions.